Bicoherence - Revision history

Admin: /* Interpretation */

2019-06-10T11:10:54Z

Interpretation

← Older revision		Revision as of 13:10, 10 June 2019
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	The two-dimensional bicoherence graph tends to show mainly two types of structures:		The two-dimensional bicoherence graph tends to show mainly two types of structures:
	* 'Points': indicating sharply defined, unchanging, locked frequencies.		* 'Points': indicating sharply defined, unchanging, locked frequencies.
	* 'Lines': these are more difficult to interpret. It is often stated that 'lines' are due to single mode (frequency) interacting with a broad range of frequencies (e.g., a [[Geodesic Acoustic Mode]] and broad-band turbulence<ref>Y. Nagashima et al, ''Observation of coherent bicoherence and biphase in potential fluctuations around geodesic acoustic mode frequency on JFT-2M'', [[doi:10.1088/0741-3335/48/5A/S38\|Plasma Phys. Control. Fusion '''48''' (2006) A377]]</ref>) - but it is not evident that this is the only explanation. Particularly, two interacting oscillators (continuously exchanging energy) ''also'' produce lines in the bicoherence graph.<ref name="milligen1995"></ref><ref>B.Ph. van Milligen, L. García, B.A. Carreras, M.A. Pedrosa, C. Hidalgo, J.A. Alonso, T. Estrada and E. Ascasíbar, MHD mode activity and the velocity shear layer at TJ-II, [[doi:10.1088/0029-5515/52/1/013006\|Nucl. Fusion 52 (2012) 013006]]</ref>		* 'Lines': these are more difficult to interpret. It is often stated that 'lines' are due to single mode (frequency) interacting with a broad range of frequencies (e.g., a [[Geodesic Acoustic Mode]] and broad-band turbulence<ref>Y. Nagashima et al, ''Observation of coherent bicoherence and biphase in potential fluctuations around geodesic acoustic mode frequency on JFT-2M'', [[doi:10.1088/0741-3335/48/5A/S38\|Plasma Phys. Control. Fusion '''48''' (2006) A377]]</ref>) - but it is not evident that this is the only explanation. Particularly, two interacting oscillators (continuously exchanging energy) ''also'' produce lines in the bicoherence graph.<ref name="milligen1995"></ref><ref>B.Ph. van Milligen, L. García, B.A. Carreras, M.A. Pedrosa, C. Hidalgo, J.A. Alonso, T. Estrada and E. Ascasíbar, ''MHD mode activity and the velocity shear layer at TJ-II'', [[doi:10.1088/0029-5515/52/1/013006\|Nucl. Fusion 52 (2012) 013006]]</ref>

	== Notes ==		== Notes ==

Admin: Corrected references

2017-02-23T09:35:54Z

Corrected references

← Older revision		Revision as of 11:35, 23 February 2017
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	The two-dimensional bicoherence graph tends to show mainly two types of structures:		The two-dimensional bicoherence graph tends to show mainly two types of structures:
	* 'Points': indicating sharply defined, unchanging, locked frequencies.		* 'Points': indicating sharply defined, unchanging, locked frequencies.
	* 'Lines': these are more difficult to interpret. It is often stated that 'lines' are due to single mode (frequency) interacting with a broad range of frequencies (e.g., a [[Geodesic Acoustic Mode]] and broad-band turbulence<ref>~~[[doi:10.1088/0741-3335/48/5A/S38\|~~Y. Nagashima et al, ''Observation of coherent bicoherence and biphase in potential fluctuations around geodesic acoustic mode frequency on JFT-2M'', Plasma Phys. Control. Fusion '''48''' (2006) A377]]</ref>) - but it is not evident that this is the only explanation. Particularly, two interacting oscillators (continuously exchanging energy) ''also'' produce lines in the bicoherence graph.<ref name="milligen1995"></ref><ref>~~[[doi:10.1088/0029-5515/52/1/013006\|~~B.Ph. van Milligen, L. García, B.A. Carreras, M.A. Pedrosa, C. Hidalgo, J.A. Alonso, T. Estrada and E. Ascasíbar, MHD mode activity and the velocity shear layer at TJ-II, Nucl. Fusion 52 (2012) 013006]]</ref>		* 'Lines': these are more difficult to interpret. It is often stated that 'lines' are due to single mode (frequency) interacting with a broad range of frequencies (e.g., a [[Geodesic Acoustic Mode]] and broad-band turbulence<ref>Y. Nagashima et al, ''Observation of coherent bicoherence and biphase in potential fluctuations around geodesic acoustic mode frequency on JFT-2M'', [[doi:10.1088/0741-3335/48/5A/S38\|Plasma Phys. Control. Fusion '''48''' (2006) A377]]</ref>) - but it is not evident that this is the only explanation. Particularly, two interacting oscillators (continuously exchanging energy) ''also'' produce lines in the bicoherence graph.<ref name="milligen1995"></ref><ref>B.Ph. van Milligen, L. García, B.A. Carreras, M.A. Pedrosa, C. Hidalgo, J.A. Alonso, T. Estrada and E. Ascasíbar, MHD mode activity and the velocity shear layer at TJ-II, [[doi:10.1088/0029-5515/52/1/013006\|Nucl. Fusion 52 (2012) 013006]]</ref>

	== Notes ==		== Notes ==

	* The bicoherence can be computed using the (continuous) wavelet transform instead of the Fourier transform, in order to improve statistics. <ref name="milligen1995">~~[http://link.aip.org/link/?PHPAEN/2/3017/1~~ B.Ph. van Milligen et al, ''Wavelet bicoherence: a new turbulence analysis tool'', Phys. Plasmas '''2''', 8 (1995) 3017]</ref>		* The bicoherence can be computed using the (continuous) wavelet transform instead of the Fourier transform, in order to improve statistics. <ref name="milligen1995">B.Ph. van Milligen et al, ''Wavelet bicoherence: a new turbulence analysis tool'', [[doi:10.1063/1.871199\|Phys. Plasmas '''2''', 8 (1995) 3017]]</ref>
	* The bicoherence can of course be defined in wavenumber space instead of frequency space by applying the replacements ''t → x'' and ''ω → k''.		* The bicoherence can of course be defined in wavenumber space instead of frequency space by applying the replacements ''t → x'' and ''ω → k''.
	* Combined temporal-spatial studies are also possible. <ref>~~[http://link.aip.org/link/PHPAEN/v17/i5/p052313/s1~~ T. Yamada, S.-I. Itoh, S. Inagaki, Y. Nagashima, S. Shinohara, N. Kasuya, K. Terasaka, K. Kamataki, H. Arakawa, M. Yagi, A. Fujisawa, and K. Itoh, ''Two-dimensional bispectral analysis of drift wave turbulence in a cylindrical plasma'' , Phys. Plasmas '''17''' (2010) 052313]</ref>		* Combined temporal-spatial studies are also possible. <ref>T. Yamada, S.-I. Itoh, S. Inagaki, Y. Nagashima, S. Shinohara, N. Kasuya, K. Terasaka, K. Kamataki, H. Arakawa, M. Yagi, A. Fujisawa, and K. Itoh, ''Two-dimensional bispectral analysis of drift wave turbulence in a cylindrical plasma'' , [[doi:10.1063/1.3429674\|Phys. Plasmas '''17''' (2010) 052313]]</ref>

	Starting from the spatio-temporal bicorrelation		Starting from the spatio-temporal bicorrelation

Admin at 09:30, 23 February 2017

2017-02-23T09:30:48Z

← Older revision		Revision as of 11:30, 23 February 2017
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	''(ω<sub>1</sub>,ω<sub>2</sub>) → (-ω<sub>1</sub>,-ω<sub>2</sub>)'', so that only one quarter of the plane ''(ω<sub>1</sub>,ω<sub>2</sub>)'' contains independent information.		''(ω<sub>1</sub>,ω<sub>2</sub>) → (-ω<sub>1</sub>,-ω<sub>2</sub>)'', so that only one quarter of the plane ''(ω<sub>1</sub>,ω<sub>2</sub>)'' contains independent information.
	Additionally, for discretely sampled data all frequencies must be less than the		Additionally, for discretely sampled data all frequencies must be less than the
	Nyquist frequency: ''\|ω<sub>1</sub>\|,\|ω<sub>2</sub>\|,\|ω\| ≤ ω<sub>Nyq</sub>''. These restrictions define a polygonal subspace of the plane, which is how the bicoherence is usually represented (for an example, see [[TJ-II:Turbulence]]).		[[wikipedia:Nyquist frequency\|Nyquist frequency]]: ''\|ω<sub>1</sub>\|,\|ω<sub>2</sub>\|,\|ω\| ≤ ω<sub>Nyq</sub>''. These restrictions define a polygonal subspace of the plane, which is how the bicoherence is usually represented (for an example, see [[TJ-II:Turbulence]]).

	The summed bicoherence is defined by		The summed bicoherence is defined by

Admin: /* Interpretation */

2012-03-30T16:32:30Z

Interpretation

← Older revision		Revision as of 18:32, 30 March 2012
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	The two-dimensional bicoherence graph tends to show mainly two types of structures:		The two-dimensional bicoherence graph tends to show mainly two types of structures:
	* 'Points': indicating sharply defined, unchanging, locked frequencies ~~(as occurs with a driving mode and a slave mode; energy transfer in a single direction)~~.		* 'Points': indicating sharply defined, unchanging, locked frequencies.
	* 'Lines': these are more difficult to interpret. It is often stated that 'lines' are due to single mode (frequency) interacting with a broad range of frequencies (e.g., a [[Geodesic Acoustic Mode]] and broad-band turbulence) - but it is not evident that this is the only explanation. Particularly, two interacting oscillators (continuously exchanging energy) ''also'' produce lines in the bicoherence graph.<ref name="milligen1995"></ref><ref>[[doi:10.1088/0029-5515/52/1/013006\|B.Ph. van Milligen, L. García, B.A. Carreras, M.A. Pedrosa, C. Hidalgo, J.A. Alonso, T. Estrada and E. Ascasíbar, MHD mode activity and the velocity shear layer at TJ-II, Nucl. Fusion 52 (2012) 013006]]</ref>		* 'Lines': these are more difficult to interpret. It is often stated that 'lines' are due to single mode (frequency) interacting with a broad range of frequencies (e.g., a [[Geodesic Acoustic Mode]] and broad-band turbulence<ref>[[doi:10.1088/0741-3335/48/5A/S38\|Y. Nagashima et al, ''Observation of coherent bicoherence and biphase in potential fluctuations around geodesic acoustic mode frequency on JFT-2M'', Plasma Phys. Control. Fusion '''48''' (2006) A377]]</ref>) - but it is not evident that this is the only explanation. Particularly, two interacting oscillators (continuously exchanging energy) ''also'' produce lines in the bicoherence graph.<ref name="milligen1995"></ref><ref>[[doi:10.1088/0029-5515/52/1/013006\|B.Ph. van Milligen, L. García, B.A. Carreras, M.A. Pedrosa, C. Hidalgo, J.A. Alonso, T. Estrada and E. Ascasíbar, MHD mode activity and the velocity shear layer at TJ-II, Nucl. Fusion 52 (2012) 013006]]</ref>

	== Notes ==		== Notes ==

Admin at 16:24, 30 March 2012

2012-03-30T16:24:44Z

← Older revision		Revision as of 18:24, 30 March 2012
Line 70:		Line 70:
	The two-dimensional bicoherence graph tends to show mainly two types of structures:		The two-dimensional bicoherence graph tends to show mainly two types of structures:
	* 'Points': indicating sharply defined, unchanging, locked frequencies (as occurs with a driving mode and a slave mode; energy transfer in a single direction).		* 'Points': indicating sharply defined, unchanging, locked frequencies (as occurs with a driving mode and a slave mode; energy transfer in a single direction).
	* 'Lines': these are more difficult to interpret. It is often stated that 'lines' are due to single mode (frequency) interacting with a broad range of frequencies (e.g., a [[Geodesic Acoustic Mode]] and broad-band turbulence) - but it is not evident that this is the only explanation. Particularly, two interacting oscillators (continuously exchanging energy) ''also'' produce lines in the bicoherence graph.		* 'Lines': these are more difficult to interpret. It is often stated that 'lines' are due to single mode (frequency) interacting with a broad range of frequencies (e.g., a [[Geodesic Acoustic Mode]] and broad-band turbulence) - but it is not evident that this is the only explanation. Particularly, two interacting oscillators (continuously exchanging energy) ''also'' produce lines in the bicoherence graph.<ref name="milligen1995"></ref><ref>[[doi:10.1088/0029-5515/52/1/013006\|B.Ph. van Milligen, L. García, B.A. Carreras, M.A. Pedrosa, C. Hidalgo, J.A. Alonso, T. Estrada and E. Ascasíbar, MHD mode activity and the velocity shear layer at TJ-II, Nucl. Fusion 52 (2012) 013006]]</ref>

	== Notes ==		== Notes ==

	* The bicoherence can be computed using the (continuous) wavelet transform instead of the Fourier transform, in order to improve statistics. <ref>[http://link.aip.org/link/?PHPAEN/2/3017/1 B.Ph. van Milligen et al, ''Wavelet bicoherence: a new turbulence analysis tool'', Phys. Plasmas '''2''', 8 (1995) 3017]</ref>		* The bicoherence can be computed using the (continuous) wavelet transform instead of the Fourier transform, in order to improve statistics. <ref name="milligen1995">[http://link.aip.org/link/?PHPAEN/2/3017/1 B.Ph. van Milligen et al, ''Wavelet bicoherence: a new turbulence analysis tool'', Phys. Plasmas '''2''', 8 (1995) 3017]</ref>
	* The bicoherence can of course be defined in wavenumber space instead of frequency space by applying the replacements ''t → x'' and ''ω → k''.		* The bicoherence can of course be defined in wavenumber space instead of frequency space by applying the replacements ''t → x'' and ''ω → k''.
	* Combined temporal-spatial studies are also possible. <ref>[http://link.aip.org/link/PHPAEN/v17/i5/p052313/s1 T. Yamada, S.-I. Itoh, S. Inagaki, Y. Nagashima, S. Shinohara, N. Kasuya, K. Terasaka, K. Kamataki, H. Arakawa, M. Yagi, A. Fujisawa, and K. Itoh, ''Two-dimensional bispectral analysis of drift wave turbulence in a cylindrical plasma'' , Phys. Plasmas '''17''' (2010) 052313]</ref>		* Combined temporal-spatial studies are also possible. <ref>[http://link.aip.org/link/PHPAEN/v17/i5/p052313/s1 T. Yamada, S.-I. Itoh, S. Inagaki, Y. Nagashima, S. Shinohara, N. Kasuya, K. Terasaka, K. Kamataki, H. Arakawa, M. Yagi, A. Fujisawa, and K. Itoh, ''Two-dimensional bispectral analysis of drift wave turbulence in a cylindrical plasma'' , Phys. Plasmas '''17''' (2010) 052313]</ref>

Admin: /* Interpretation */

2012-03-30T16:18:06Z

Interpretation

← Older revision		Revision as of 18:18, 30 March 2012
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	the phase between the wave at ω and the sum wave		the phase between the wave at ω and the sum wave
	ω<sub>1</sub>+ω<sub>2</sub> is nearly constant over a significant number of realizations.		ω<sub>1</sub>+ω<sub>2</sub> is nearly constant over a significant number of realizations.

			The two-dimensional bicoherence graph tends to show mainly two types of structures:
			* 'Points': indicating sharply defined, unchanging, locked frequencies (as occurs with a driving mode and a slave mode; energy transfer in a single direction).
			* 'Lines': these are more difficult to interpret. It is often stated that 'lines' are due to single mode (frequency) interacting with a broad range of frequencies (e.g., a [[Geodesic Acoustic Mode]] and broad-band turbulence) - but it is not evident that this is the only explanation. Particularly, two interacting oscillators (continuously exchanging energy) ''also'' produce lines in the bicoherence graph.

	== Notes ==		== Notes ==

Admin at 13:41, 6 July 2011

2011-07-06T13:41:40Z

← Older revision		Revision as of 15:41, 6 July 2011
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	== Bispectrum ==		== Bispectrum ==

	~~Denoting the~~ Fourier transforms of the signals ''X<sub>i</sub>(t)'' by		The Fourier transforms of the signals ''X<sub>i</sub>(t)'' are denoted by

	:<math>\hat X_i(\omega)</math>		:<math>\hat X_i(\omega)</math>

	the bispectrum ~~is defined~~ as		Thus, the bispectrum, computed as the Fourier transform of the bicorrelation ''C<sub>2</sub>'', becomes:

	:<math>B(\omega_1,\omega_2) = \hat X_1^*(\omega)\hat X_2(\omega_1) \hat X_2(\omega_2)</math>		:<math>B(\omega_1,\omega_2) = \hat X_1^*(\omega)\hat X_2(\omega_1) \hat X_2(\omega_2)</math>
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	:<math>\omega = \omega_1 + \omega_2</math>		:<math>\omega = \omega_1 + \omega_2</math>

			Hence, the bispectrum is interpreted as a measure of the degree of three-wave coupling.

	== Bicoherence ==		== Bicoherence ==

Admin at 19:03, 28 September 2010

2010-09-28T19:03:42Z

← Older revision		Revision as of 21:03, 28 September 2010
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	== Notes ==		== Notes ==

	* The bicoherence can of course be defined in wavenumber space instead of frequency space by applying the replacements ''t → x'' and ''ω → k''. Combined temporal-spatial studies are also possible. <ref>[http://link.aip.org/link/PHPAEN/v17/i5/p052313/s1 T. Yamada, S.-I. Itoh, S. Inagaki, Y. Nagashima, S. Shinohara, N. Kasuya, K. Terasaka, K. Kamataki, H. Arakawa, M. Yagi, A. Fujisawa, and K. Itoh, ''Two-dimensional bispectral analysis of drift wave turbulence in a cylindrical plasma'' , Phys. Plasmas '''17''' (2010) 052313]</ref>
	* The bicoherence can be computed using the (continuous) wavelet transform instead of the Fourier transform, in order to improve statistics. <ref>[http://link.aip.org/link/?PHPAEN/2/3017/1 B.Ph. van Milligen et al, ''Wavelet bicoherence: a new turbulence analysis tool'', Phys. Plasmas '''2''', 8 (1995) 3017]</ref>		* The bicoherence can be computed using the (continuous) wavelet transform instead of the Fourier transform, in order to improve statistics. <ref>[http://link.aip.org/link/?PHPAEN/2/3017/1 B.Ph. van Milligen et al, ''Wavelet bicoherence: a new turbulence analysis tool'', Phys. Plasmas '''2''', 8 (1995) 3017]</ref>
			* The bicoherence can of course be defined in wavenumber space instead of frequency space by applying the replacements ''t → x'' and ''ω → k''.
			* Combined temporal-spatial studies are also possible. <ref>[http://link.aip.org/link/PHPAEN/v17/i5/p052313/s1 T. Yamada, S.-I. Itoh, S. Inagaki, Y. Nagashima, S. Shinohara, N. Kasuya, K. Terasaka, K. Kamataki, H. Arakawa, M. Yagi, A. Fujisawa, and K. Itoh, ''Two-dimensional bispectral analysis of drift wave turbulence in a cylindrical plasma'' , Phys. Plasmas '''17''' (2010) 052313]</ref>

			Starting from the spatio-temporal bicorrelation

			:<math>C_{22}(x_1,x_2,t_1,t_2) = \left \langle X_1(x,t)X_2(x+x_1,t+t_1)X_2(x+x_2,t+t_2) \right \rangle </math>

			the spatio-temporal bispectrum is

			:<math>B_2(k_1,k_2,\omega_1,\omega_2) = \hat X_1^*(k,\omega)\hat X_2(k_1,\omega_1) \hat X_2(k_2,\omega_2)</math>

			where <math>\omega = \omega_1 + \omega_2</math> and <math>k=k_1+k_2</math>.

	== References ==		== References ==
	<references />		<references />

Admin: /* Notes */

2010-09-26T16:07:55Z

Notes

← Older revision		Revision as of 18:07, 26 September 2010
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	== Notes ==		== Notes ==

	* The bicoherence can of course be defined in wavenumber space instead of frequency space by applying the replacements ''t → x'' and ''ω → k''.		* The bicoherence can of course be defined in wavenumber space instead of frequency space by applying the replacements ''t → x'' and ''ω → k''. Combined temporal-spatial studies are also possible. <ref>[http://link.aip.org/link/PHPAEN/v17/i5/p052313/s1 T. Yamada, S.-I. Itoh, S. Inagaki, Y. Nagashima, S. Shinohara, N. Kasuya, K. Terasaka, K. Kamataki, H. Arakawa, M. Yagi, A. Fujisawa, and K. Itoh, ''Two-dimensional bispectral analysis of drift wave turbulence in a cylindrical plasma'' , Phys. Plasmas '''17''' (2010) 052313]</ref>
	* The bicoherence can be computed using the (continuous) wavelet transform instead of the Fourier transform, in order to improve statistics. <ref>[http://link.aip.org/link/?PHPAEN/2/3017/1 B.Ph. van Milligen et al, ''Wavelet bicoherence: a new turbulence analysis tool'', Phys. Plasmas '''2''', 8 (1995) 3017]</ref>		* The bicoherence can be computed using the (continuous) wavelet transform instead of the Fourier transform, in order to improve statistics. <ref>[http://link.aip.org/link/?PHPAEN/2/3017/1 B.Ph. van Milligen et al, ''Wavelet bicoherence: a new turbulence analysis tool'', Phys. Plasmas '''2''', 8 (1995) 3017]</ref>

	== References ==		== References ==
	<references />		<references />

Admin at 13:32, 25 September 2010

2010-09-25T13:32:33Z

← Older revision		Revision as of 15:32, 25 September 2010
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	:<math>X_i(t)\,</math>		:<math>X_i(t)\,</math>

			For convenience and simplicity of notation, the data can be taken to have
			''zero mean'' (<math>\langle X_i \rangle = 0</math>) and
			''unit standard deviation'' (<math>\langle X_i^2 \rangle = 1</math>).

	The standard cross spectrum is the Fourier transform of the correlation		The standard cross spectrum is the Fourier transform of the correlation